Orthogonal Projection is Mapping

Theorem

Let $H$ be a Hilbert space.

Let $K$ be a closed linear subspace of $H$.

Let $P_K: H \to H$ be the orthogonal projection on $K$.

Then $P_K$ is a mapping.

Proof

For $P_K$ to be a mapping we need to show that:

$\forall h \in H: \map{P_K} h$ exists and is unique

By definition of $\map{P_K} h$, this amounts to:

There is a unique $k \in K$ such that $\norm{ h - k } = \map d {h, K}$

This is precisely the statement of Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space.

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality